Draw straight lines for sphere/polar coordinate meshes
Here's a view of a sphere from directly above one of the poles:

The orange selected vertices look like a square, don't they?  that's because I used the knife tool to carefully draw &quot;straight&quot; lines between four of the vertices.
But let's look at the sphere from a slightly different view:

Notice how in 3d the &quot;Square&quot; doesn't have right angles or straight lines?
Next I added a seam around the equator of the sphere.  (It's the red line that's outlining the equator in both drawings.  In the UV editor, I selected all vertices and did a UV unwrap.  Here's what you get for two hemispheres:

Notice that the apparent square is curved?
When you want to UV uwrap a sphere and draw in a box you end up with this sort of distortion.  Rather than all of the quadrants being separate, I would settle for two hemispheres, I think.
To draw in the boxes, I would export the UV map and then import it into an editor to use as a guide layer for my texture painting.
Here is one way to do that:
In the UV editor, type ALT-N to bring up the new image menu:

Give the image a name, and then make sure you've selected all of the vertices in the sphere.
You should end up with an image that looks something like this:

Next export the image to a file by typing SHIFT-ALT-S and using the dialog box.  Take that image to your favorite drawing program and make it a layer.  Do your drawing on it.  Then, before you save the drawing, remove the layer containing the image.  That'll be your texture file.